Python linalg.svd函数代码示例

    def normalise_site_svd(self, site, left=True):
        '''
        Normalise the MP at a given site. Use the SVD method,
        arXiv:1008.3477, sec. 4.1.3
        '''
        if (left and site == L) or (not left and site == 1):
            self.normalise_extremal_site(site, left=left)
            return
        
        M_asb = self.mps[site] # the matrix we have to work on, M_site
        r, s, c = M_asb.shape  # row, spin, column

        if left:
            M_aab = M_asb.reshape((r * s, c)) # merge first two indices

            U, S, Vh = la.svd(M_aab, full_matrices=False)
            self.mps[site] = U.reshape((r, s, U.size // (r * s))) # new A_site

            # Contract over index b: (S @ Vh)_ab (X) (M_site+1)_bsc
            SVh = np.dot(S, Vh)
            self.mps[site+1] = np.tensordot(SVh, self.mps[site+1], ((1),(0)))
        else:
            M_aab = M_asb.reshape((r, s * c)) # merge last two indices

            U, S, Vh = la.svd(M_aab, full_matrices=False)
            self.mps[site] = Vh.reshape((Vh.size // (s * c), s, c)) # new B
            # Contract over index b: (M_site-1)_asb (X) (U @ S)_bc 
            US = np.dot(U, S)
            self.mps[site-1] = np.tensordot(self.mps[site-1], US, ((2),(0)))
        return

    def mixed_canonicalise(site):
        '''
        Put the MPS in mixed canonical form.

        Sites 1...site are left-normalised, sites site...L are 
        right-normalised. The singular values are stored in self.S and 
        returned.
        '''
        if site > self.left_normalised:
            # gotta left-normalise sites left_normalised...site
            # left qr up to site-1
            for _s in range(self.left_normalised+1, site):
                self.normalise_site_qr(_s, left=True)
            # svd at site
            Msite = self.mps[site]
            r, s, c = Msite.shape
            Msite = Msite.reshape((r * s, c))
            U, self.S, Vh = la.svd(Msite, full_matrices=False)
            self.mps[site] = U.reshape((r, s, U.size // (r * s))) # new A_site
            self.mps[site+1] = np.tensordot(Vh, self.mps[site+1], ((1),(0))) 
        else:
            # gotta right-normalise sites right_normalised...site+1
            # right qr up to site+2
            for _s in range(self.right_normalised, site+1):
                self.normalise_site_qr(_s, left=False)
            # svd at site+1
            Msite = self.mps[site+1]
            r, s, c = Msite.shape
            Msite = Msite.reshape((r, s * c))
            U, self.S, Vh = la.svd(Msite, full_matrices=False)
            self.mps[site+1] = Vh.reshape((Vh // (s * c), s,c)) #new B
            self.mps[site] = np.tensordot(self.mps[site], U, ((1),(0))) 

        return self.S

def ideal_data(num, dimU, dimY, dimX, noise=1):
    """Linear system data"""
    # generate randomized linear system matrices
    A = randn(dimX, dimX)
    B = randn(dimX, dimU)
    C = randn(dimY, dimX)
    D = randn(dimY, dimU)

    # make sure state evolution is stable
    U, S, V = svd(A)
    A = dot(U, dot(diag(S / max(S)), V))
    U, S, V = svd(B)
    S2 = zeros((size(U,1), size(V,0)))
    S2[:,:size(U,1)] = diag(S / max(S))
    B = dot(U, dot(S2, V))

    # random input
    U = randn(num, dimU)

    # initial state
    X = reshape(randn(dimX), (1,-1))

    # initial output
    Y = reshape(dot(C, X[-1]) + dot(D, U[0]), (1,-1))

    # generate next state
    X = concatenate((X, reshape(dot(A, X[-1]) + dot(B, U[0]), (1,-1))))

    # and so forth
    for u in U[1:]:
        Y = concatenate((Y, reshape(dot(C, X[-1]) + dot(D, u), (1,-1))))
        X = concatenate((X, reshape(dot(A, X[-1]) + dot(B, u), (1,-1))))

    return U, Y + randn(num, dimY) * noise

def randomized_svd(A, rank=None, power_iter=2, k=50, p=50, block_krylov=True, l=0, use_id=False):
    """
    :param rank: If None, adaptive range finder is used to detect the rank.
    :param power_iter: Number of power iterations.
    :param k: Initial estimate of rank, if adaptive range finder is used.
    :param p: Batch size in the incremental steps of adaptive range finder.
    :param l: Number of overestimated columns in computing Q.
    """
    if rank is None:
        Q = adaptive_range(A, k=k, p=p, power_iter_k=power_iter)
    else:
        Q = randomized_range(A, rank, power_iter=power_iter, block_krylov=block_krylov, l=l)

    if rank is None:
        rank = Q.shape[1]

    if use_id:
        J, P = ID_row(Q, rank)
        W, R = la.qr(A[J, :].T, mode='economic')
        Q = P.dot(R.T)
        U, sigma, VT = la.svd(Q, full_matrices=False)
        V = VT.dot(W.T)
        return U[:, :rank], sigma[:rank], V[:rank, :]

    M = Q.T.dot(A)
    U, sigma, VT = la.svd(M, full_matrices=False)
    U = Q.dot(U)
    return U[:, :rank], sigma[:rank], VT[:rank, :]

def _compute_subcorr(G, phi_sig):
    """Compute the subspace correlation."""
    Ug, Sg, Vg = linalg.svd(G, full_matrices=False)
    tmp = np.dot(Ug.T.conjugate(), phi_sig)
    Uc, Sc, Vc = linalg.svd(tmp, full_matrices=False)
    X = np.dot(np.dot(Vg.T, np.diag(1. / Sg)), Uc)  # subcorr
    return Sc[0], X[:, 0] / linalg.norm(X[:, 0])

def compute_bench(data_gen, samples_range, features_range, q=3):

    it = 0

    results = defaultdict(lambda: [])

    max_it = len(samples_range) * len(features_range)
    for n_samples in samples_range:
        for n_features in features_range:
            it += 1
            print '===================='
            print 'Iteration %03d of %03d' % (it, max_it)
            print '===================='
            X = make_data(n_samples, n_features)
            rank = min(n_samples, n_samples) / 10 + 1

            gc.collect()
            print "benching scipy svd: "
            tstart = time()
            svd(X, full_matrices=False)
            results['scipy svd'].append(time() - tstart)

            gc.collect()
            print "benching scikit-learn fast_svd: q=0"
            tstart = time()
            fast_svd(X, rank, q=0)
            results['scikit-learn fast_svd (q=0)'].append(time() - tstart)

            gc.collect()
            print "benching scikit-learn fast_svd: q=%d " % q
            tstart = time()
            fast_svd(X, rank, q=q)
            results['scikit-learn fast_svd (q=%d)' % q].append(time() - tstart)

    return results

    def setdata(self, X, V):
        A = self.bialtprodeye(2*self.F.J_coords)
        """Note: p, q <= min(n,m)"""

        self.data.Brand = 2*(random((A.shape[0],self.data.p))-0.5)
        self.data.Crand = 2*(random((A.shape[1],self.data.q))-0.5)
        self.data.B = zeros((A.shape[0],self.data.p), Float)
        self.data.C = zeros((A.shape[1],self.data.q), Float)
        self.data.D = zeros((self.data.q,self.data.p), Float)

        U, S, Vh = linalg.svd(A)
        self.data.b = U[:,-1:]
        self.data.c = num_transpose(Vh)[:,-1:]
        
        if self.update:
            self.data.B[:,1] = self.data.b
            self.data.C[:,1] = self.data.c
            
            U2, S2, Vh2 = linalg.svd(c_[r_[A, transpose(self.data.C[:,1])], r_[self.data.B[:,1], [[0]]]])
            self.data.B[:,2] = U2[0:A.shape[0],-1:]
            self.data.C[:,2] = num_transpose(Vh2)[0:A.shape[1],-1:]
            self.data.D[0,1] = U2[A.shape[0],-1]
            self.data.D[1,0] = num_transpose(Vh2)[A.shape[1],-1]
        else:
            self.data.B = self.data.Brand
            self.data.C = self.data.Crand

    def __init__(self,data,cov_matrix=False,loc=None):
        """Parameters
        ----------
        data : array of data, shape=(number points,number dim)
               If cov_matrix is True then data is the covariance matrix (see below)

        Keywords
        --------
        cov_matrix : bool (optional)
                     If True data is treated as a covariance matrix with shape=(number dim, number dim)
        loc        : the mean of the data if a covarinace matrix is given, shape=(number dim)
        """
        if cov_matrix:
            self.dim=data.shape[0]
            self.n=None
            self.data_t=None
            self.mu=loc
            self.evec,eval,V=sl.svd(data,full_matrices=False)
            self.sigma=sqrt(eval)
            self.Sigma=diag(1./self.sigma)
            self.B=dot(self.evec,self.Sigma)
            self.Binv=sl.inv(self.B)
        else:
            self.n,self.dim=data.shape #the shape of input data
            self.mu=data.mean(axis=0) #the mean of the data
            self.data_t=data-self.mu #remove the mean
            self.evec,eval,V=sl.svd(self.data_t.T,full_matrices=False) #get the eigenvectors (axes of the ellipsoid)
            data_p=dot(self.data_t,self.evec) #project the data onto the eigenvectors
            self.sigma=data_p.std(axis=0) #get the spread of the distribution (the axis ratos for the ellipsoid)
            self.Sigma=diag(1./self.sigma) #the eigenvalue matrix for the ellipsoid equation
            self.B=dot(self.evec,self.Sigma) #used in the ellipsoid equation
            self.Binv=sl.inv(self.B) #also useful to have around

def rcca(X, Y, reg):   
    X = np.array(X)
    Y = np.array(Y)
    n, p = X.shape
    n, q = Y.shape
        
    # zero mean
    X = X - X.mean(axis=0)
    Y = Y - Y.mean(axis=0)
        
    # covariances
    S = np.cov(X.T, Y.T, bias=1)
    
    SXX = S[:p,:p]
    SYY = S[p:,p:]
    SXY = S[:p,p:]

    ux, lx, vxt = SLA.svd(SXX)
    #uy, ly, vyt = SLA.svd(SYY)
    
    #print lx
    #正則化
    Rg = np.diag(np.ones(p)*reg)
    SXX = SXX + Rg
    SYY = SYY + Rg

    sqx = SLA.sqrtm(SLA.inv(SXX)) # SXX^(-1/2)
    sqy = SLA.sqrtm(SLA.inv(SYY)) # SYY^(-1/2)
    M = np.dot(np.dot(sqx, SXY), sqy.T) # SXX^(-1/2) * SXY * SYY^(-T/2)
    A, r, Bh = SLA.svd(M, full_matrices=False)
    B = Bh.T      
    #r = self.reg*r
    return r[0], lx[0], lx[0] 

def compute_fundamental(x1,x2):
  """ 正規化8点法を使って対応点群(x1,x2:3*nの配列)
      から基礎行列を計算する。各列は次のような並びである。
      [x'*x, x'*y, x', y'*x, y'*y, y', x, y, 1] """

  n = x1.shape[1]
  if x2.shape[1] != n:
    raise ValueError("Number of points don't match.")

  # 方程式の行列を作成する
  A = zeros((n,9))
  for i in range(n):
    A[i] = [x1[0,i]*x2[0,i], x1[0,i]*x2[1,i], x1[0,i]*x2[2,i],
        x1[1,i]*x2[0,i], x1[1,i]*x2[1,i], x1[1,i]*x2[2,i],
        x1[2,i]*x2[0,i], x1[2,i]*x2[1,i], x1[2,i]*x2[2,i] ]

  # 線形最小2乗法で計算する
  U,S,V = linalg.svd(A)
  F = V[-1].reshape(3,3)

  # Fの制約
  # 最後の特異値を0にして階数を2にする
  U,S,V = linalg.svd(F)
  S[2] = 0
  F = dot(U,dot(diag(S),V))

  return F

    def dataNorm(self):
        SXX = np.cov(self.X)
        U, l, Ut = LA.svd(SXX, full_matrices=True) 
        H = np.dot(LA.sqrtm(LA.inv(np.diag(l))),Ut)
        self.nX = np.dot(H,self.X)

        #print np.cov(self.nX)
        #print "mean:"
        #print np.mean(self.nX)

        SYY = np.cov(self.Y)
        U, l, Ut = LA.svd(SYY, full_matrices=True) 
        H = np.dot(LA.sqrtm(LA.inv(np.diag(l))),Ut)
        #print "H"
        #print H
        self.nY = np.dot(H,self.Y)
        #print np.cov(self.nY)

        print "dataNorm_X:"
        for i in range(len(self.nX)):
            print(self.nX[i])
        print("---")

        print "dataNorm_Y:"
        for i in range(len(self.nY)):
            print(self.nY[i])
        print("---")

def geigen(Amat, Bmat, Cmat):
    """
    generalized eigenvalue problem of the form

    max tr L'AM / sqrt(tr L'BL tr M'CM) w.r.t. L and M

    :param Amat numpy ndarray of shape (M,N)
    :param Bmat numpy ndarray of shape (M,N)
    :param Bmat numpy ndarray of shape (M,N)

    :rtype: numpy ndarray
    :return values: eigenvalues
    :return Lmat: left eigenvectors
    :return Mmat: right eigenvectors

    """
    if Bmat.shape[0] != Bmat.shape[1]:
        print("BMAT is not square.\n")
        sys.exit(1)

    if Cmat.shape[0] != Cmat.shape[1]:
        print("CMAT is not square.\n")
        sys.exit(1)

    p = Bmat.shape[0]
    q = Cmat.shape[0]

    s = min(p, q)
    tmp = fabs(Bmat - Bmat.transpose())
    tmp1 = fabs(Bmat)
    if tmp.max() / tmp1.max() > 1e-10:
        print("BMAT not symmetric..\n")
        sys.exit(1)

    tmp = fabs(Cmat - Cmat.transpose())
    tmp1 = fabs(Cmat)
    if tmp.max() / tmp1.max() > 1e-10:
        print("CMAT not symmetric..\n")
        sys.exit(1)

    Bmat = (Bmat + Bmat.transpose()) / 2.
    Cmat = (Cmat + Cmat.transpose()) / 2.
    Bfac = cholesky(Bmat)
    Cfac = cholesky(Cmat)
    Bfacinv = inv(Bfac)
    Bfacinvt = Bfacinv.transpose()
    Cfacinv = inv(Cfac)
    Dmat = Bfacinvt.dot(Amat).dot(Cfacinv)
    if p >= q:
        u, d, v = svd(Dmat)
        values = d
        Lmat = Bfacinv.dot(u)
        Mmat = Cfacinv.dot(v.transpose())
    else:
        u, d, v = svd(Dmat.transpose())
        values = d
        Lmat = Bfacinv.dot(u)
        Mmat = Cfacinv.dot(v.transpose())

    return values, Lmat, Mmat

def  addblock_svd_update( Uarg, Sarg, Varg, Aarg, force_orth = False):
  U = Varg
  V = Uarg
  S = np.eye(len(Sarg),len(Sarg))*Sarg
  A = Aarg.T
  
  current_rank = U.shape[1]
  m = np.dot(U.T,A)
  p = A - np.dot(U,m)
  P = lin.orth(p)
  Ra = np.dot(P.T,p)
  z = np.zeros(m.shape)
  K = np.vstack(( np.hstack((S,m)), np.hstack((z.T,Ra)) ))
  tUp,tSp,tVp = lin.svd(K);
  tUp = tUp[:,:current_rank]
  tSp = np.diag(tSp[:current_rank])
  tVp = tVp[:,:current_rank]
  Sp = tSp
  Up = np.dot(np.hstack((U,P)),tUp)
  Vp = np.dot(V,tVp[:current_rank,:])
  Vp = np.vstack((Vp, tVp[current_rank:tVp.shape[0], :]))
  
  if force_orth:
    UQ,UR = lin.qr(Up,mode='economic')
    VQ,VR = lin.qr(Vp,mode='economic')
    tUp,tSp,tVp = lin.svd( np.dot(np.dot(UR,Sp),VR.T));
    tSp = np.diag(tSp)
    Up = np.dot(UQ,tUp)
    Vp = np.dot(VQ,tVp)
    Sp = tSp;

  Up1 = Vp;
  Vp1 = Up;
    
  return Up1,Sp,Vp1

def compute_fundamental(x1,x2):
    """    Computes the fundamental matrix from corresponding points 
        (x1,x2 3*n arrays) using the 8 point algorithm.
        Each row in the A matrix below is constructed as
        [x'*x, x'*y, x', y'*x, y'*y, y', x, y, 1] """
    
    n = x1.shape[1]
    if x2.shape[1] != n:
        raise ValueError("Number of points don't match.")
    
    # build matrix for equations
    A = zeros((n,9))
    for i in range(n):
        A[i] = [x1[0,i]*x2[0,i], x1[0,i]*x2[1,i], x1[0,i]*x2[2,i],
                x1[1,i]*x2[0,i], x1[1,i]*x2[1,i], x1[1,i]*x2[2,i],
                x1[2,i]*x2[0,i], x1[2,i]*x2[1,i], x1[2,i]*x2[2,i] ]
            
    # compute linear least square solution
    U,S,V = linalg.svd(A)
    F = V[-1].reshape(3,3)
        
    # constrain F
    # make rank 2 by zeroing out last singular value
    U,S,V = linalg.svd(F)
    S[2] = 0
    F = dot(U,dot(diag(S),V))
    
    return F/F[2,2]

def rpca(M, eps=0.001, r=1):
    '''
    An implementation of the robust pca algorithm as
    outlined in [need reference]
    '''
    assert(len(M.shape) == 2)

    m,n = M.shape

    s = linalg.svd(M, compute_uv=False)
    B = 1 / np.sqrt(n) # threshold parameter
    thresh = B * s[0]

    # Initial Low Rank Component
    L = np.zeros(M.shape)
    # Initial Sparse Component
    S = HT(M - L, thresh)

    iterations = range(int(10 * np.log(n * B * frob_norm(M - S) / eps)))
    logger.info('Number of iterations: %d to achieve eps = %f' % (len(iterations), eps))

    for k in xrange(1, r+1):
        for t in iterations:

            U,s,Vt = linalg.svd(M - S, full_matrices=False)
            thresh = B * ( s[k] + s[k-1] * (1/2)**t )

            # Best rank k approximation of M - S
            L = np.dot(np.dot(U[:,:k], np.diag(s[:k])), Vt[:k])
            S = HT(M - L, thresh)

        if (B * s[k]) < (eps / (2*n)):
            break

    return L,S